In a large bowl, whisk together the flour, granulated sugar, salt, and active dry yeast.
Pour the warm milk into the dry ingredients, then add the egg, melted butter, and vanilla extract.
Mix the ingredients until combined, then knead the dough for 8-10 minutes until it becomes smooth and elastic.
Cover the dough and let it rise in a warm place for about 1 hour, or until doubled in size.
On a lightly floured surface, roll out the dough to about 1 cm thickness. Use a glass or a doughnut cutter to cut out circles.
Place the cut-out doughnuts on a baking sheet lined with parchment paper, and cover them with a clean kitchen towel. Let them rise for another 30 minutes.
In a large pot, heat the oil to about 340°F (170°C). Test the oil by dropping in a doughnut; if it immediately floats to the surface and gently bubbles, the oil is ready.
Fry the doughnuts for 1-2 minutes per side, until golden brown. Use a slotted spoon to remove them, and place them on paper towels to drain excess oil.
Once the doughnuts have cooled slightly, use a piping bag to fill them with salmonberry jam.
Dust the doughnuts with powdered sugar and a pinch of cinnamon for an extra burst of flavor.
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